Seki in Go

Authors: Vladimir GURVICH, Andrey GOL'BERG

Group Analysis of Seki Positions

Firstly consider the following three problems:
The questions are the same: Is the game over?

A seki position will be called complete if any move
by either player leads to some loss for him. What
kinds of complete seki exist? For example two
eyeless groups and two dame between them
(dia. 1);
or two one-eyed groups and one dame between them
(dia. 2);
or one eyeless White group, two one-eyed
Black groups, and two dame
(dia. 3).
Something else?
Using one's imaginaion, more abstract examples can be
constructed. Such as a ring of arbitrary size,
consisting of alternating eyeless Black and White
groups, with one dame between each pair of
neighbouring groups. Another example can be obtained
by breaking up this ring, and substituting the end
groups by one-eyed ones.

In this paper we consider some examples, some hypotheses,
and results connected with a classification of complete
seki. It is natural to classify them according to
the number of groups and the number of dame between them.
We first consider the case of eyeless groups.

Let us define the seki-matrix. Its rows correspond to
Black groups, and its columns correspond to White groups.
At the intersection of a row and a column we put the
number of dame between the corresponding groups. See the
diagrams --
dias. 0, 1-3,
dias. 4-9,
dias. 10-12, and
dias. 13, and 14
-- for examples.

We now have a new game using matrices with integer
non-negative elements. The players (Black and White)
move in turn. A "move" consists of subtracting 1 from
any positive element. Black wins if, after his move, a
column of zeroes appears. Similarly, White wins if,
after his move, a row of zeroes appears. If both a
zero-column, and a zero-row, appear simultaneously after a
move, the player of that move wins.

This game will be called "Nil".

The analogy with Go is clear. A move in Nil corresponds
to a move in Go where a stone is put on a dame. A
zero-row corresponds to a captured Black group and a
zero-column corresponds to a captured White group. A
matrix in Nil such that any move by either player leads
to a loss for them corresponds to a complete seki.
However exclusions [? exceptions] are possible, as we
shall see later. Such matrices will be called seki
matrices.

The 1x1 matrix M0 = (2) gives the simplest example (see
dia. 1).
There exists no 2x1 seki matrix. This means
that three eyeless groups can't form a seki. There are
four 2x2 seki matrices:

corresponds to the same seki situation as M2. Generally
we will not distinguish between matrices which differ
only by a permutation of rows or columns. Note also
that the matrix M4 corresponds to two independent copies
of the matrix of type M0
(dia. 1).
More generally, any
matrix that can be transformed to the block-diagonal
form (see
dia. 7)
by permutation of the rows and columns
actually corresponds to some set of independent positions
in Go. Such matrices will be called disconnected. We
will consider connected matrices only.

The matrix:

M5 = (3 2)
(2 3)

is already not a seki-matrix. Really the corresponding
seki in Go can be completed -- see
dia. 8.

Let us state some results and hypotheses concerning
eyeless sekis.

Hypothesis 1: All seki-matrices are even-sum-squares
-- that is the number of rows is equal to the number of
columns, and the sum of every row and every column is a
constant. This sum will be called the index of an
even-sum-square. In the language of Go an index is the
number of dame, provided this number is the same for all
groups involved.

Theorem: If the matrix is an even-sum-square, and its
index equals 2, 3, or 4 then the player who moves first
can't win in Nil. In other words, the corresponding
positions in Go are sekis (complete or incomplete).
However the theorem is not valid for indices greater
than 4 -- see the example in
dia. 9 --
all the groups
there have 5 dame, but if Black plays first he captures
all the White stones.

Hypothesis 2: For any positive integer "n", there
exists only a finite number of nxn seki-matrices.

The game "Nil" can easily be changed so as to describe
sekis which include one-eyed groups. For this purpose,
let us add to the matrix one special column (consisting
of zeros and ones) indicating the number of eyes of the
Black groups. Similarly, add a special row indicating
the number of eyes of the White groups. Now Black
(White) has first to obtain a zero column (row), and
then possibly play one extra move to remove the eye
(if there is one).

Examples are given in
dias. 2 and 3. Note that for both
of them, the analogy to Hypothesis 1 is valid -- that is
the sum in all the rows and columns (excluding the
additional special ones) is a constant. However there
are examples which contradict this hypothesis -- see
dias. 10 and 11.

Until now, in our study of sekis, we have assumed that
if a player loses one of his groups then he loses.
However there are positions in Go which contradict this
seemingly natural statement. Consider, for example,
dia. 12.
The matrix shown there is not a seki-matrix
-- actually White can capture the Black group in the
upper right corner with 2 moves, however this manouevre
in a net loss for White since he loses the larger White
group in the lower left corner. Therefore the position in
dia. 12
is a complete seki.

Now consider another example -- dia. 13.
The matrix
shown there is a seki matrix. One can easily prove that
the player who moves first loses the Nil game. This
means literally that the player who moves first in
the game of Go is the first one to lose a group.
However, both Black and White profit by sacrificing
one of their groups, because after that it is then
possible to capture the more important opponent's
group. Therefore this position is not complete.
If Black moves first he wins by 2 points; if White
moves first the game ends in Jigo.

These examples show that an exchange of groups is
possible in seki-type positions. If lots of exchanges
are possible, it may be rather difficult to decide whether
a position is complete -- see, for example,
dia. 14.

To reduce the dimension of
dia. 14,
we dared to consider
Go fields without the centre point. Why not? One may
play Go on an arbitrary graph and the problems considered
here will be the same.

Moscow 25-3-1981
Transcription by Harry Fearnley of a paper by Vladimir Gurvich and
Andrey Gol'berg snd sent to Prof Klaus Heine -- from a poor quality photocopy.